\(\int (d \cos (a+b x))^{9/2} \csc (a+b x) \, dx\) [222]

Optimal result

Integrand size = 19, antiderivative size = 100 \[ \int (d \cos (a+b x))^{9/2} \csc (a+b x) \, dx=\frac {d^{9/2} \arctan \left (\frac {\sqrt {d \cos (a+b x)}}{\sqrt {d}}\right )}{b}-\frac {d^{9/2} \text {arctanh}\left (\frac {\sqrt {d \cos (a+b x)}}{\sqrt {d}}\right )}{b}+\frac {2 d^3 (d \cos (a+b x))^{3/2}}{3 b}+\frac {2 d (d \cos (a+b x))^{7/2}}{7 b} \]

[Out]

d^(9/2)*arctan((d*cos(b*x+a))^(1/2)/d^(1/2))/b-d^(9/2)*arctanh((d*cos(b*x+a))^(1/2)/d^(1/2))/b+2/3*d^3*(d*cos(
b*x+a))^(3/2)/b+2/7*d*(d*cos(b*x+a))^(7/2)/b

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {2645, 327, 335, 304, 209, 212} \[ \int (d \cos (a+b x))^{9/2} \csc (a+b x) \, dx=\frac {d^{9/2} \arctan \left (\frac {\sqrt {d \cos (a+b x)}}{\sqrt {d}}\right )}{b}-\frac {d^{9/2} \text {arctanh}\left (\frac {\sqrt {d \cos (a+b x)}}{\sqrt {d}}\right )}{b}+\frac {2 d^3 (d \cos (a+b x))^{3/2}}{3 b}+\frac {2 d (d \cos (a+b x))^{7/2}}{7 b} \]

[In]

Int[(d*Cos[a + b*x])^(9/2)*Csc[a + b*x],x]

[Out]

(d^(9/2)*ArcTan[Sqrt[d*Cos[a + b*x]]/Sqrt[d]])/b - (d^(9/2)*ArcTanh[Sqrt[d*Cos[a + b*x]]/Sqrt[d]])/b + (2*d^3*
(d*Cos[a + b*x])^(3/2))/(3*b) + (2*d*(d*Cos[a + b*x])^(7/2))/(7*b)

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 304

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}
, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !
GtQ[a/b, 0]

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 2645

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[-(a*f)^(-1), Subst[
Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2]
&&  !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[m, n])

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {x^{9/2}}{1-\frac {x^2}{d^2}} \, dx,x,d \cos (a+b x)\right )}{b d} \\ & = \frac {2 d (d \cos (a+b x))^{7/2}}{7 b}-\frac {d \text {Subst}\left (\int \frac {x^{5/2}}{1-\frac {x^2}{d^2}} \, dx,x,d \cos (a+b x)\right )}{b} \\ & = \frac {2 d^3 (d \cos (a+b x))^{3/2}}{3 b}+\frac {2 d (d \cos (a+b x))^{7/2}}{7 b}-\frac {d^3 \text {Subst}\left (\int \frac {\sqrt {x}}{1-\frac {x^2}{d^2}} \, dx,x,d \cos (a+b x)\right )}{b} \\ & = \frac {2 d^3 (d \cos (a+b x))^{3/2}}{3 b}+\frac {2 d (d \cos (a+b x))^{7/2}}{7 b}-\frac {\left (2 d^3\right ) \text {Subst}\left (\int \frac {x^2}{1-\frac {x^4}{d^2}} \, dx,x,\sqrt {d \cos (a+b x)}\right )}{b} \\ & = \frac {2 d^3 (d \cos (a+b x))^{3/2}}{3 b}+\frac {2 d (d \cos (a+b x))^{7/2}}{7 b}-\frac {d^5 \text {Subst}\left (\int \frac {1}{d-x^2} \, dx,x,\sqrt {d \cos (a+b x)}\right )}{b}+\frac {d^5 \text {Subst}\left (\int \frac {1}{d+x^2} \, dx,x,\sqrt {d \cos (a+b x)}\right )}{b} \\ & = \frac {d^{9/2} \arctan \left (\frac {\sqrt {d \cos (a+b x)}}{\sqrt {d}}\right )}{b}-\frac {d^{9/2} \text {arctanh}\left (\frac {\sqrt {d \cos (a+b x)}}{\sqrt {d}}\right )}{b}+\frac {2 d^3 (d \cos (a+b x))^{3/2}}{3 b}+\frac {2 d (d \cos (a+b x))^{7/2}}{7 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.83 \[ \int (d \cos (a+b x))^{9/2} \csc (a+b x) \, dx=\frac {d^4 \sqrt {d \cos (a+b x)} \left (21 \arctan \left (\sqrt {\cos (a+b x)}\right )-21 \text {arctanh}\left (\sqrt {\cos (a+b x)}\right )+2 \cos ^{\frac {3}{2}}(a+b x) \left (7+3 \cos ^2(a+b x)\right )\right )}{21 b \sqrt {\cos (a+b x)}} \]

[In]

Integrate[(d*Cos[a + b*x])^(9/2)*Csc[a + b*x],x]

[Out]

(d^4*Sqrt[d*Cos[a + b*x]]*(21*ArcTan[Sqrt[Cos[a + b*x]]] - 21*ArcTanh[Sqrt[Cos[a + b*x]]] + 2*Cos[a + b*x]^(3/
2)*(7 + 3*Cos[a + b*x]^2)))/(21*b*Sqrt[Cos[a + b*x]])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(330\) vs. \(2(80)=160\).

Time = 0.40 (sec) , antiderivative size = 331, normalized size of antiderivative = 3.31

method	result	size
default	\(-\frac {96 d^{4} \sqrt {-2 d \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+d}\, \sqrt {-d}\, \left (\sin ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+21 d^{\frac {9}{2}} \ln \left (-\frac {2 \left (2 d \cos \left (\frac {b x}{2}+\frac {a}{2}\right )-\sqrt {d}\, \sqrt {-2 d \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+d}+d \right )}{\cos \left (\frac {b x}{2}+\frac {a}{2}\right )+1}\right ) \sqrt {-d}+21 d^{\frac {9}{2}} \ln \left (\frac {4 d \cos \left (\frac {b x}{2}+\frac {a}{2}\right )+2 \sqrt {d}\, \sqrt {-2 d \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+d}-2 d}{\cos \left (\frac {b x}{2}+\frac {a}{2}\right )-1}\right ) \sqrt {-d}-144 d^{4} \sqrt {-2 d \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+d}\, \sqrt {-d}\, \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+128 d^{4} \sqrt {-2 d \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+d}\, \sqrt {-d}\, \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-40 d^{4} \sqrt {-2 d \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+d}\, \sqrt {-d}+42 d^{5} \ln \left (\frac {2 \sqrt {-d}\, \sqrt {-2 d \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+d}-2 d}{\cos \left (\frac {b x}{2}+\frac {a}{2}\right )}\right )}{42 \sqrt {-d}\, b}\)	\(331\)

[In]

int((d*cos(b*x+a))^(9/2)*csc(b*x+a),x,method=_RETURNVERBOSE)

[Out]

-1/42/(-d)^(1/2)*(96*d^4*(-2*d*sin(1/2*b*x+1/2*a)^2+d)^(1/2)*(-d)^(1/2)*sin(1/2*b*x+1/2*a)^6+21*d^(9/2)*ln(-2/
(cos(1/2*b*x+1/2*a)+1)*(2*d*cos(1/2*b*x+1/2*a)-d^(1/2)*(-2*d*sin(1/2*b*x+1/2*a)^2+d)^(1/2)+d))*(-d)^(1/2)+21*d
^(9/2)*ln(2/(cos(1/2*b*x+1/2*a)-1)*(2*d*cos(1/2*b*x+1/2*a)+d^(1/2)*(-2*d*sin(1/2*b*x+1/2*a)^2+d)^(1/2)-d))*(-d
)^(1/2)-144*d^4*(-2*d*sin(1/2*b*x+1/2*a)^2+d)^(1/2)*(-d)^(1/2)*sin(1/2*b*x+1/2*a)^4+128*d^4*(-2*d*sin(1/2*b*x+
1/2*a)^2+d)^(1/2)*(-d)^(1/2)*sin(1/2*b*x+1/2*a)^2-40*d^4*(-2*d*sin(1/2*b*x+1/2*a)^2+d)^(1/2)*(-d)^(1/2)+42*d^5
*ln(2/cos(1/2*b*x+1/2*a)*((-d)^(1/2)*(-2*d*sin(1/2*b*x+1/2*a)^2+d)^(1/2)-d)))/b

Fricas [A] (verification not implemented)

none

Time = 0.39 (sec) , antiderivative size = 313, normalized size of antiderivative = 3.13 \[ \int (d \cos (a+b x))^{9/2} \csc (a+b x) \, dx=\left [\frac {42 \, \sqrt {-d} d^{4} \arctan \left (\frac {2 \, \sqrt {d \cos \left (b x + a\right )} \sqrt {-d}}{d \cos \left (b x + a\right ) + d}\right ) + 21 \, \sqrt {-d} d^{4} \log \left (-\frac {d \cos \left (b x + a\right )^{2} + 4 \, \sqrt {d \cos \left (b x + a\right )} \sqrt {-d} {\left (\cos \left (b x + a\right ) - 1\right )} - 6 \, d \cos \left (b x + a\right ) + d}{\cos \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) + 1}\right ) + 8 \, {\left (3 \, d^{4} \cos \left (b x + a\right )^{3} + 7 \, d^{4} \cos \left (b x + a\right )\right )} \sqrt {d \cos \left (b x + a\right )}}{84 \, b}, -\frac {42 \, d^{\frac {9}{2}} \arctan \left (\frac {2 \, \sqrt {d \cos \left (b x + a\right )} \sqrt {d}}{d \cos \left (b x + a\right ) - d}\right ) - 21 \, d^{\frac {9}{2}} \log \left (-\frac {d \cos \left (b x + a\right )^{2} - 4 \, \sqrt {d \cos \left (b x + a\right )} \sqrt {d} {\left (\cos \left (b x + a\right ) + 1\right )} + 6 \, d \cos \left (b x + a\right ) + d}{\cos \left (b x + a\right )^{2} - 2 \, \cos \left (b x + a\right ) + 1}\right ) - 8 \, {\left (3 \, d^{4} \cos \left (b x + a\right )^{3} + 7 \, d^{4} \cos \left (b x + a\right )\right )} \sqrt {d \cos \left (b x + a\right )}}{84 \, b}\right ] \]

[In]

integrate((d*cos(b*x+a))^(9/2)*csc(b*x+a),x, algorithm="fricas")

[Out]

[1/84*(42*sqrt(-d)*d^4*arctan(2*sqrt(d*cos(b*x + a))*sqrt(-d)/(d*cos(b*x + a) + d)) + 21*sqrt(-d)*d^4*log(-(d*
cos(b*x + a)^2 + 4*sqrt(d*cos(b*x + a))*sqrt(-d)*(cos(b*x + a) - 1) - 6*d*cos(b*x + a) + d)/(cos(b*x + a)^2 +
2*cos(b*x + a) + 1)) + 8*(3*d^4*cos(b*x + a)^3 + 7*d^4*cos(b*x + a))*sqrt(d*cos(b*x + a)))/b, -1/84*(42*d^(9/2
)*arctan(2*sqrt(d*cos(b*x + a))*sqrt(d)/(d*cos(b*x + a) - d)) - 21*d^(9/2)*log(-(d*cos(b*x + a)^2 - 4*sqrt(d*c
os(b*x + a))*sqrt(d)*(cos(b*x + a) + 1) + 6*d*cos(b*x + a) + d)/(cos(b*x + a)^2 - 2*cos(b*x + a) + 1)) - 8*(3*
d^4*cos(b*x + a)^3 + 7*d^4*cos(b*x + a))*sqrt(d*cos(b*x + a)))/b]

Sympy [F(-1)]

Timed out. \[ \int (d \cos (a+b x))^{9/2} \csc (a+b x) \, dx=\text {Timed out} \]

[In]

integrate((d*cos(b*x+a))**(9/2)*csc(b*x+a),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.98 \[ \int (d \cos (a+b x))^{9/2} \csc (a+b x) \, dx=\frac {42 \, d^{\frac {11}{2}} \arctan \left (\frac {\sqrt {d \cos \left (b x + a\right )}}{\sqrt {d}}\right ) + 21 \, d^{\frac {11}{2}} \log \left (\frac {\sqrt {d \cos \left (b x + a\right )} - \sqrt {d}}{\sqrt {d \cos \left (b x + a\right )} + \sqrt {d}}\right ) + 12 \, \left (d \cos \left (b x + a\right )\right )^{\frac {7}{2}} d^{2} + 28 \, \left (d \cos \left (b x + a\right )\right )^{\frac {3}{2}} d^{4}}{42 \, b d} \]

[In]

integrate((d*cos(b*x+a))^(9/2)*csc(b*x+a),x, algorithm="maxima")

[Out]

1/42*(42*d^(11/2)*arctan(sqrt(d*cos(b*x + a))/sqrt(d)) + 21*d^(11/2)*log((sqrt(d*cos(b*x + a)) - sqrt(d))/(sqr
t(d*cos(b*x + a)) + sqrt(d))) + 12*(d*cos(b*x + a))^(7/2)*d^2 + 28*(d*cos(b*x + a))^(3/2)*d^4)/(b*d)

Giac [F]

\[ \int (d \cos (a+b x))^{9/2} \csc (a+b x) \, dx=\int { \left (d \cos \left (b x + a\right )\right )^{\frac {9}{2}} \csc \left (b x + a\right ) \,d x } \]

[In]

integrate((d*cos(b*x+a))^(9/2)*csc(b*x+a),x, algorithm="giac")

[Out]

integrate((d*cos(b*x + a))^(9/2)*csc(b*x + a), x)

Mupad [F(-1)]

Timed out. \[ \int (d \cos (a+b x))^{9/2} \csc (a+b x) \, dx=\int \frac {{\left (d\,\cos \left (a+b\,x\right )\right )}^{9/2}}{\sin \left (a+b\,x\right )} \,d x \]

[In]

int((d*cos(a + b*x))^(9/2)/sin(a + b*x),x)

[Out]

int((d*cos(a + b*x))^(9/2)/sin(a + b*x), x)